Finding Independent Sets inK4-Free 4-Regular Connected Graphs

LetGbe aK4-free 4-regular connected graph withνvertices. ThenGcontains an independent set of vertices of cardinality at least (7ν−4)/26. Furthermore, the proof will yield a polynomial-time algorithm which will return an independent set of cardinality at least (7ν−4)/26 for any such graph. A family ofK4-free 4-regular connected graphs with maximum independent set of cardinality311νis provided, so that the asymptotically sharp lower bound lies between726and311. We also provide a slight improvement over the best previously known result for triangle-free 5-regular graphs.